Download 1. Define rational expressions. 2. Define rational functions and give

6.1
Rational Expressions and Functions; Multiplying and Dividing
1.
2.
3.
4.
5.
6.
Define rational expressions.
Define rational functions and give their domains.
Write rational expressions in lowest terms.
Multiply rational expressions.
Find reciprocals of rational expressions.
Divide rational expressions.
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
1
Objective 1
Define rational expressions.
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
2
Define rational expressions.
In arithmetic, a rational number is the quotient of two
integers, with the denominator not 0. In algebra, a
rational expression, or algebraic fraction, is the quotient
of two polynomials, again with the denominator not 0.
5

x −a m + 4 8 x − 2 x + 5
x 
5
,
,
,
, and x  or

2
y 4 m−2
4 x + 5x
1


2
Rational expressions are elements of the set
P

 P and Q are polynomials, where Q ≠ 0 .
Q

Copyright © 2016, 2012, 2008 Pearson Education, Inc.
3
Objective 2
Define rational functions and give
their domains.
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
4
Define rational functions and give their
domains.
A rational function is defined by a quotient of
polynomials and has the form
P( x)
f ( x)
, where Q( x) ≠ 0.
=
Q( x)
The domain of a rational function includes all real
numbers except those that make Q(x)—that is, the
denominator—equal to 0.
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
5
Classroom
Example 1
Finding Domains of Rational
Functions
Give the domain of each rational function using setbuilder notation and interval notation.
a. f ( x) = x + 6
x2 − x − 6
Values of x that make the denominator 0 must be
excluded from the domain. To find these values, set
the denominator equal to 0 and solve the resulting
equation. x 2 − x − 6 =
0 { x | x is a real number, x ≠ −2,3}
( x + 2)( x − 3) =
0
( −∞, − 2 ) ∪ ( −2, 3) ∪ ( 3, ∞ )
=
x + 2 0 or =
x−3 0
x=
−2 or x =
3
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
6
Classroom
Example 1
Finding Domains of Rational
Functions (cont.)
Give the domain of each rational function using setbuilder notation and interval notation.
b. f ( x) = 3 + 2 x
5
The denominator, 5, cannot ever be 0, so the domain
includes all real numbers (−∞, ∞).
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
7
Objective 3
Write rational expressions in lowest
terms.
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
8
Fundamental Property of Rational
Numbers
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
9
Writing a Rational Expression in
Lowest Terms
Step 1 Factor both numerator and denominator to find
their greatest common factor (GCF).
Step 2 Apply the fundamental property. Divide out
common factors.
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
10
Classroom
Example 2
Writing Rational Expressions in
Lowest Terms
Write each rational expression in lowest terms.
y2 + 2 y − 3
a. 2
y − 3y + 2
1 + p3
c.
1+ p
( y + 3)( y − 1)
=
( y − 2)( y − 1)
(1 + p )(1 − p + p )
=
1+ p
y+3
=
y−2
=1 − p + p 2
2
y + 2 The denominator cannot be factored, so
b. 2
y + 4 this expression cannot be simplified
further and is in lowest terms.
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
11
Writing Rational Expressions in
Lowest Terms
Classroom
Example 3
Write each rational expression in lowest terms.
p−2
b.
4 − p2
r −1
a.
1− r
r −1
=
(−1)(r − 1)
p−2
=
−1( p 2 − 4)
1
=
= −1
−1
p−2
=
−1( p − 2)( p + 2)
1
1
=
=−
(−1)( p + 2)
p+2
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
12
Quotient of Opposites
In general, if the numerator and the denominator of a
rational expression are opposites, then the expression
equals –1.
Based on this result, the following are true statements.
However, the following expression cannot be simplified
further.
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
13
Objective 4
Multiply rational expressions.
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
14
Multiply Rational Expressions
Step 1 Factor all numerators and denominators as
completely as possible.
Step 2 Apply the fundamental property.
Step 3 Multiply remaining factors in the numerators
and remaining factors in the denominators.
Leave the denominator in factored form.
Step 4 Check to be sure the product is in lowest terms.
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
15
Classroom
Example 4
Multiplying Rational Expressions
Multiply.
2
2
a. c + 2c c − 4c + 4
⋅
2
c −4
c2 − c
c(c + 2)
(c − 2)(c − 2)
c−2
=
⋅
=
(c + 2)(c − 2)
c(c − 1)
c −1
m 2 − 16 1
b.
⋅
m+2 m+4
(m + 4)(m − 4) 1
m−4
⋅
=
m+2
m+4
m+2
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
16
Objective 5
Find reciprocals of rational
expressions.
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
17
Finding the Reciprocal
When dividing rational numbers, we use reciprocals.
The reciprocal of a rational expression is defined in the
same way. Two rational expressions are reciprocals of
each other if they have a product of 1.
Finding the Reciprocal
To find the reciprocal of a nonzero rational expression,
interchange the numerator and denominator of the
expression.
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
18
Objective 6
Divide rational expressions.
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
19
Dividing Rational Expressions
To divide two rational expressions, multiply the first
expression (the dividend) by the reciprocal of the
second expression (the divisor).
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
20
Classroom
Example 5
Dividing Rational Expressions
Divide.
5 p + 2 15 p + 6
a.
÷
6
5
q 2 + 2q 4 − q 2
b. 5 + q ÷ 3q − 6
2
q
+ 2q 3q − 6
5p + 2
5
=
⋅
2
=
⋅
5+q 4−q
6
15 p + 6
q (q + 2)
3(q − 2)
=
⋅
5p + 2
5
5 + q (2 − q )(2 + q )
=
⋅
6
3(5 p + 2)
3(q − 2)
q (q + 2)
=
⋅
5 + q (−1)(q − 2)(q + 2)
5
=
3q
18
= −
5+q
Copyright © 2016, 2012, 2008 Pearson Education, Inc.
21